program snbdr6 c c ... Construct matrices A and M in LAPACK-style band form. c The matrix A is a block tridiagonal matrix. Each c diagonal block is a tridiagonal matrix with c 4 on the diagonal, 1-rho*h/2 on the subdiagonal and c 1+rho*h/2 on the superdiagonal. Each subdiagonal block c of A is an identity matrix. The matrix M is the c tridiagonal matrix with 4 on the diagonal and 1 on the c subdiagonal and superdiagonal. c c ... Define COMPLEX shift SIGMA=(SIGMAR,SIGMAI), SIGMAI .ne. zero. c c ... Call snband to find eigenvalues LAMBDA closest to SIGMA c such that c A*x = LAMBDA*M*x. c c ... Use mode 4 of SNAUPD. c c\BeginLib c c\Routines called: c snband ARPACK banded eigenproblem solver. c slapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully. c slaset LAPACK routine to initialize a matrix to zero. c saxpy Level 1 BLAS that computes y <- alpha*x+y. c snrm2 Level 1 BLAS that computes the norm of a vector. c sgbmv Level 2 BLAS that computes the band matrix vector product. c c\Author c Danny Sorensen c Richard Lehoucq c Chao Yang c Dept. of Computational & c Applied Mathematics c Rice University c Houston, Texas c c\SCCS Information: @(#) c FILE: nbdr6.F SID: 2.5 DATE OF SID: 08/26/96 RELEASE: 2 c c\Remarks c 1. None c c\EndLib c c--------------------------------------------------------------------- c c %-------------------------------------% c | Define leading dimensions for all | c | arrays. | c | MAXN - Maximum size of the matrix | c | MAXNEV - Maximum number of | c | eigenvalues to be computed | c | MAXNCV - Maximum number of Arnoldi | c | vectors stored | c | MAXBDW - Maximum bandwidth | c %-------------------------------------% c integer maxn, maxnev, maxncv, maxbdw, lda, & lworkl, ldv parameter ( maxn = 1000, maxnev = 25, maxncv=50, & maxbdw=50, lda = maxbdw, ldv = maxn ) c c %--------------% c | Local Arrays | c %--------------% c integer iparam(11), iwork(maxn) logical select(maxncv) Real & a(lda,maxn), m(lda,maxn), rfac(lda,maxn), & workl(3*maxncv*maxncv+6*maxncv), workd(3*maxn), & workev(3*maxncv), v(ldv, maxncv), & resid(maxn), d(maxncv, 3), ax(maxn), mx(maxn) Complex & cfac(lda, maxn), workc(maxn) c c %---------------% c | Local Scalars | c %---------------% c character which*2, bmat integer nev, ncv, ku, kl, info, i, j, ido, & n, nx, lo, idiag, isup, isub, mode, maxitr, & nconv logical rvec, first Real & tol, rho, h, sigmar, sigmai c c %------------% c | Parameters | c %------------% c Real & one, zero, two parameter (one = 1.0E+0 , zero = 0.0E+0 , & two = 2.0E+0 ) c c %--------------------% c | Intrinsic function | c %--------------------% c intrinsic abs c c %-----------------------------% c | BLAS & LAPACK routines used | c %-----------------------------% c Real & slapy2, snrm2 external slapy2, snrm2, sgbmv, saxpy c c %-----------------------% c | Executable Statements | c %-----------------------% c c %-----------------------------------------------------% c | The number NX is the size of each diagonal block of | c | A. The number N(=NX*NX) is the dimension of the | c | matrix. The number N(=NX*NX) is the dimension of | c | the matrix. A generalized eigenvalue problem is | c | solved (BMAT = 'G'). NEV numbers of eigenvalues | c | closest to the COMPLEX shift (SIGMAR,SIGMAI) | c | (WHICH='LM') and their corresponding eigenvectors | c | are computed. The user can modify NX, NEV, NCV, | c | WHICH to solve problems of different sizes, and | c | to get different parts the spectrum. However, the | c | following rules must be satisfied: | c | N <= MAXN | c | NEV <= MAXNEV | c | NEV + 2 <= NCV <= MAXNCV | c %-----------------------------------------------------% c nx = 10 n = nx*nx nev = 4 ncv = 10 if ( n .gt. maxn ) then print *, ' ERROR with _NBDR6: N is greater than MAXN ' go to 9000 else if ( nev .gt. maxnev ) then print *, ' ERROR with _NBDR6: NEV is greater than MAXNEV ' go to 9000 else if ( ncv .gt. maxncv ) then print *, ' ERROR with _NBDR6: NCV is greater than MAXNCV ' go to 9000 end if bmat = 'G' which = 'LM' sigmar = 4.0E-1 sigmai = 6.0E-1 c c %-----------------------------------------------------% c | The work array WORKL is used in SNAUPD as | c | workspace. Its dimension LWORKL is set as | c | illustrated below. The parameter TOL determines | c | the stopping criterion. If TOL<=0, machine | c | precision is used. The variable IDO is used for | c | reverse communication, and is initially set to 0. | c | Setting INFO=0 indicates that a random vector is | c | generated in SNAUPD to start the Arnoldi iteration. | c %-----------------------------------------------------% c lworkl = 3*ncv**2+6*ncv tol = zero ido = 0 info = 0 c c %---------------------------------------------------% c | IPARAM(3) specifies the maximum number of Arnoldi | c | iterations allowed. Mode 4 of SNAUPD is used | c | (IPARAm(7) = 4). All these options can be changed | c | by the user. For details, see the documentation | c | in snband. | c %---------------------------------------------------% c maxitr = 300 mode = 4 c iparam(3) = maxitr iparam(7) = mode c c %--------------------------------------------% c | Construct matrices A and M in LAPACK-style | c | banded form. | c %--------------------------------------------% c c %---------------------------------------------% c | Zero out the workspace for banded matrices. | c %---------------------------------------------% c call slaset('A', lda, n, zero, zero, a, lda) call slaset('A', lda, n, zero, zero, m, lda) call slaset('A', lda, n, zero, zero, rfac, lda) c c %-------------------------------------% c | KU, KL are number of superdiagonals | c | and subdiagonals within the band of | c | matrices A and M. | c %-------------------------------------% c kl = nx ku = nx c c %---------------% c | Main diagonal | c %---------------% c idiag = kl+ku+1 do 30 j = 1, n a(idiag,j) = 4.0E+0 m(idiag,j) = 4.0E+0 30 continue c c %-------------------------------------% c | First subdiagonal and superdiagonal | c %-------------------------------------% c isup = kl+ku isub = kl+ku+2 h = one / real (nx+1) rho = 1.0E+2 do 50 i = 1, nx lo = (i-1)*nx do 40 j = lo+1, lo+nx-1 a(isub,j+1) = -one+h*rho/two a(isup,j) = -one-h*rho/two 40 continue 50 continue c do 60 j = 1, n-1 m(isub,j+1) = one m(isup,j) = one 60 continue c c %------------------------------------% c | KL-th subdiagonal and KU-th super- | c | diagonal. | c %------------------------------------% c isup = kl+1 isub = 2*kl+ku+1 do 80 i = 1, nx-1 lo = (i-1)*nx do 70 j = lo+1, lo+nx a(isup,nx+j) = -one a(isub,j) = -one 70 continue 80 continue c c %------------------------------------------------% c | Call ARPACK banded solver to find eigenvalues | c | and eigenvectors. The real parts of the | c | eigenvalues are returned in the first column | c | of D, the imaginary parts are returned in the | c | second column of D. Eigenvectors are returned | c | in the first NCONV (=IPARAM(5)) columns of V. | c %------------------------------------------------% c rvec = .true. call snband(rvec, 'A', select, d, d(1,2), v, ldv, sigmar, & sigmai, workev, n, a, m, lda, rfac, cfac, ku, kl, & which, bmat, nev, tol, resid, ncv, v, ldv, iparam, & workd, workl, lworkl, workc, iwork, info) c if ( info .eq. 0) then c c %-----------------------------------% c | Print out convergence information | c %-----------------------------------% c nconv = iparam(5) c print *, ' ' print *, ' _NBDR6 ' print *, ' ====== ' print *, ' ' print *, ' The size of the matrix is ', n print *, ' Number of eigenvalue requested is ', nev print *, ' The number of Arnoldi vectors generated', & ' (NCV) is ', ncv print *, ' The number of converged Ritz values is ', & nconv print *, ' What portion of the spectrum ', which print *, ' The number of Implicit Arnoldi ', & ' update taken is ', iparam(3) print *, ' The number of OP*x is ', iparam(9) print *, ' The convergence tolerance is ', tol print *, ' ' c c %----------------------------% c | Compute the residual norm. | c | || A*x - lambda*x || | c %----------------------------% c first = .true. do 90 j = 1, nconv c if ( d(j,2) .eq. zero ) then c c %--------------------% c | Ritz value is real | c %--------------------% c call sgbmv('Notranspose', n, n, kl, ku, one, & a(kl+1,1), lda, v(1,j), 1, zero, & ax, 1) call sgbmv('Notranspose', n, n, kl, ku, one, & m(kl+1,1), lda, v(1,j), 1, zero, & mx, 1) call saxpy(n, -d(j,1), mx, 1, ax, 1) d(j,3) = snrm2(n, ax, 1) d(j,3) = d(j,3) / abs(d(j,1)) c else if ( first ) then c c %------------------------% c | Ritz value is complex | c | Residual of one Ritz | c | value of the conjugate | c | pair is computed. | c %------------------------% c call sgbmv('Notranspose', n, n, kl, ku, one, & a(kl+1,1), lda, v(1,j), 1, zero, & ax, 1) call sgbmv('Notranspose', n, n, kl, ku, one, & m(kl+1,1), lda, v(1,j), 1, zero, & mx, 1) call saxpy(n, -d(j,1), mx, 1, ax, 1) call sgbmv('Notranspose', n, n, kl, ku, one, & m(kl+1,1), lda, v(1,j+1), 1, zero, & mx, 1) call saxpy(n, d(j,2), mx, 1, ax, 1) d(j,3) = snrm2(n, ax, 1) call sgbmv('Notranspose', n, n, kl, ku, one, & a(kl+1,1), lda, v(1,j+1), 1, zero, & ax, 1) call sgbmv('Notranspose', n, n, kl, ku, one, & m(kl+1,1), lda, v(1,j+1), 1, zero, & mx, 1) call saxpy(n, -d(j,1), mx, 1, ax, 1) call sgbmv('Notranspose', n, n, kl, ku, one, & m(kl+1,1), lda, v(1,j), 1, zero, & mx, 1) call saxpy(n, -d(j,2), mx, 1, ax, 1) d(j,3) = slapy2( d(j,3), snrm2(n, ax, 1) ) d(j,3) = d(j,3) / slapy2(d(j,1),d(j,2)) d(j+1,3) = d(j,3) first = .false. else first = .true. end if c 90 continue call smout(6, nconv, 3, d, maxncv, -6, & 'Ritz values (Real,Imag) and relative residuals') else c c %-------------------------------------% c | Either convergence failed, or there | c | is error. Check the documentation | c | for snband. | c %-------------------------------------% c print *, ' ' print *, ' Error with _nband, info= ', info print *, ' Check the documentation of _nband ' print *, ' ' c end if c 9000 end